---
title: "How missing hydrogen atoms revealed the Lamb shift"
description: "How did a dip in a downstream signal reveal energy missing from a theory of hydrogen?"
canonical_url: "https://sandboxphysics.com/stories/quantum-evidence/lamb-shift-microwave-evidence/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/lamb-shift-microwave-evidence/"
last_modified: "2026-10-24"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# How missing hydrogen atoms revealed the Lamb shift

Sandbox Physics · 2026-10-15

How did a dip in a downstream signal reveal energy missing from a theory of hydrogen?

## Two levels that should have coincided

In 1947, Lamb and Retherford used microwaves to investigate hydrogen and found a small separation between two excited energy levels that theory placed together. The S level was above the P level by an amount corresponding to roughly one gigahertz, or a billion microwave oscillations per second.

The result posed a precise problem for the successful Dirac theory of hydrogen. It explained important details of the fine structure but gave these two states the same energy. The experiment focused on that particular relationship.

Read the evidence: W. E. Lamb Jr. and R. C. Retherford (1947), Fine Structure of the Hydrogen Atom by a Microwave Method Lamb and Retherford (1947), original paper scan hosted by the University of Maryland, pp. 241–243

## Let long-lived atoms report a transition

Some atoms in the beam occupied a metastable state: an excited state that survived long enough to reach a downstream target. On striking the metal, these atoms released detectable electrons and produced a current.

A related excited state had a much shorter lifetime. If a microwave transferred an atom into it, the atom quickly decayed and no longer reached the target as a metastable atom. The current fell. The researchers had converted a hard-to-see internal change into a count of surviving atoms.

The initial report held the microwave frequency fixed and varied the magnetic field. The field changed the atomic level spacing until the transition matched the microwaves. A dip in downstream current marked the conditions where atoms were readily transferred out of the long-lived state.

Read the evidence: Lamb and Retherford (1947), original paper scan hosted by the University of Maryland, pp. 241–243

## Infer the separation without a magnetic field

The observed transitions changed with the magnetic field. Their pattern allowed the researchers to infer the separation of the two levels at zero field. The equation defines the usual positive Lamb shift. The state labels identify the excited S and P levels with total electronic angular momentum one half; h is Planck’s constant, which converts frequency into energy. The relevant structures are treated separately in this definition.

The original paper reported an approximate one-gigahertz separation. Its figure caption explicitly said that a translated comparison curve was not the best fit. The short-lived state’s natural width, hyperfine structure, strong microwave excitation and field inhomogeneity all broadened the signal. Field calibration also limited the precision.

$$
h\nu_{\mathrm L}=E(2S_{1/2})-E(2P_{1/2})
$$

Read the evidence: Lamb and Retherford (1947), original paper scan hosted by the University of Maryland, pp. 241–243

## Include the atom’s electromagnetic interaction

Bethe responded in the same year with a calculation of electromagnetic energy shifts. An electron in an atom also interacts with the electromagnetic field. Effects absent from the ideal Dirac–Coulomb problem became visible in this small energy difference.

The result helped drive the development of quantum electrodynamics and set a target for increasingly precise comparisons. Later experiments and calculations would account for smaller corrections. The clear achievement in 1947 was to separate two levels that should have coincided: a current dip pointed directly to a relationship the theory needed to repair.

Read the evidence: H. A. Bethe (1947), The Electromagnetic Shift of Energy Levels, Phys. Rev. 72, 339 Lamb and Retherford (1947), original paper scan hosted by the University of Maryland, pp. 241–243

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [Find a microwave resonance in simulated surviving-atom counts](https://sandboxphysics.com/labs/lamb-shift-microwave/)
- [Compare the hydrogen energy levels before the Lamb correction](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/)

## Sources and reading notes

We read the 1947 Lamb–Retherford paper, including the signal method and figure captions, in a scan held by the University of Maryland. Its approximate one-gigahertz result is kept apart from later precision values. Bethe’s paper is a theoretical follow-up, not a second experiment.

1. [W. E. Lamb Jr. and R. C. Retherford (1947), Fine Structure of the Hydrogen Atom by a Microwave Method](https://doi.org/10.1103/PhysRev.72.241)
2. [Lamb and Retherford (1947), original paper scan hosted by the University of Maryland, pp. 241–243](https://physics.umd.edu/grt/taj/624b/Lamb1947.pdf)
3. [H. A. Bethe (1947), The Electromagnetic Shift of Energy Levels, Phys. Rev. 72, 339](https://doi.org/10.1103/PhysRev.72.339)

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